If $f$is an odd function then ${\int }_{-1}^1\left[\left|x\right|+f\left(x\right)\cos x\right]dx=$

The function $\left|x\right|$is an even function & $f\left(x\right)\cos x$is an odd function 

$\therefore {\int }_{-1}^1\left[\left|x\right|+f\left(x\right)\cos x\right]dx=2{\int }_0^1\left|x\right|dx+0=2{\int }_0^{1}xdx=1$